Communications on Applied Nonlinear Analysis 

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Bipolar Vague 𝜶 Generalized Continuous Mappings in Topological 

Spaces 

 

F. Prishka1 and Dr. L. Mariapresenti2 

1Research Scholar, Department of Mathematics, Nirmala College for Women, Redfield’s, Coimbatore, Tamil Nadu, India.  

2Assistant Professor, Department of Mathematics, Nirmala College for Women, Redfield’s, Coimbatore, Tamil Nadu, 

India. 

Email1: prishkamaths@gmail.com  and Email2: presentimaria88@gmail.com 

 

Article History: 

Received: 10-06-2024 

Revised: 10-07-2024 

Accepted: 30-07-2024 

Abstract:  

In this paper we have introduced bipolar vague 𝛼 generalized continuous mappings in 

topological spaces and investigated some of their properties. Also, we have provided some 

characterization of bipolar vague 𝛼 generalized continuous mappings in topological spaces. 

Keywords: Bipolar vague sets, bipolar vague topology, bipolar vague 𝛼 generalized closed 

sets, bipolar vague 𝛼 generalized continuous mappings and bipolar vague 𝛼 generalized 

irresolute mappings. 

 

1. Introduction 

Fuzzy set was introduced by L.A.Zadeh [11] in 1965. The concept of fuzzy topology was introduced 

by C.L.Chang [3] in 1968. The generalized closed sets in general topology were first introduced by 

N.Levine [9] in 1970. K.Atanassov [2] in 1986 introduced the concept of intuitionistic fuzzy sets. The 

notion of vague set theory was introduced by W.L.Gau and D.J.Buehrer [7] in 1993. D.Coker [6] in 

1997 introduced intuitionistic fuzzy topological spaces. Bipolar- valued fuzzy sets, which was 

introduced by K.M.Lee [8] in 2000 is an extension of fuzzy sets whose membership degree range is 

enlarged from the interval [0, 1] to [-1,1]. A new class of generalized bipolar vague sets was introduced 

by S.Cicily Flora and I.Arockiarani [4] in 2016. F.Prishka and L.Mariapresenti [10] introduced bipolar 

vague 𝛼 generalized closed sets in topological spaces. In continuation of our research work we have 

introduced bipolar vague 𝛼 generalized continuous mappings in topological spaces, bipolar vague 𝛼 

generalized irresolute mappings in topological spaces and investigated some of their properties. Also, 

we have provided some characterization of bipolar vague 𝛼 generalized continuous mappings in 

topological spaces. 

2. Preliminaries 

Here in this paper the bipolar vague topological spaces are denoted by (X, B𝑉𝜏). Also, the bipolar 

vague interior, bipolar vague closure of a bipolar vague set A are denoted by BVInt(A) and BVCl(A). 

The complement of a bipolar vague set A is denoted by Ac and the empty set and whole sets are denoted 

by 0~ and  1~ respectively. 

Definition 2.1: [8] Let X be the universe. Then a bipolar valued fuzzy sets, A on X is defined by 

positive membership function 𝜇𝐴
+, that is 𝜇𝐴

+: X→ [0,1], and a negative membership function 𝜇𝐴
−, that 



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is  𝜇𝐴
−: X→ [-1,0]. For the sake of simplicity, we shall use the symbol                                                      A 

= {𝑥, 𝜇𝐴
+(𝑥), 𝜇𝐴

−(x): 𝑥 ∈ 𝑋}. 

Definition 2.2: [8] Let A and B be two bipolar valued fuzzy sets then their union, intersection and 

complement are defined as follows: 

(i) 𝜇𝐴∪𝐵
+ = max {𝜇𝐴

+(𝑥), 𝜇𝐵
+𝑥)} 

(ii) 𝜇𝐴∪𝐵
− = min {𝜇𝐴

−(𝑥), 𝜇𝐵
−𝑥)} 

(iii) 𝜇𝐴∩𝐵
+ = min {𝜇𝐴

+(𝑥), 𝜇𝐵
+𝑥)} 

(iv) 𝜇𝐴∩𝐵
− = max {𝜇𝐴

−(𝑥), 𝜇𝐵
−𝑥)} 

(v) 𝜇𝐴𝑐
+ (x) = 1-𝜇𝐴

+(𝑥) and 𝜇𝐴𝑐
− (x) = -1-𝜇𝐴

−(𝑥) for all 𝑥 ∈ 𝑋. 

Definition 2.3: [7] A vague set A in the universe of discourse U is a pair of (𝑡𝐴, 𝑓𝐴) where                     𝑡𝐴: 

U→[0,1], 𝑓𝐴: U→[0,1] are the mapping such that 𝑡𝐴 + 𝑓𝐴 ≤ 1 for all 𝑢 ∈ 𝑈. The function 𝑡𝐴 and 𝑓𝐴 are 

called true membership function and false membership function respectively. The interval [𝑡𝐴, 1 − 𝑓𝐴] 

is called the vague value of u in A, and denoted by 𝜈𝐴(𝑢), that is                                               𝜈𝐴(𝑢) = 

[𝑡𝐴(𝑢), 1 − 𝑓(𝑢)]. 

Definition 2.4: [7] Let A be a non-empty set and the vague set A and B in the form                                                    

A = {𝑥, 𝑡𝐴(𝑥), 1 − 𝑓𝐴(𝑥): x ∈ X }, B = {𝑥, 𝑡𝐵(𝑥), 1 − 𝑓𝐵(𝑥): x ∈ X }. Then 

(i) A ⊆ B if and only if  𝑡𝐴(𝑥)  ≤   𝑡𝐵(𝑥) and 1 − 𝑓𝐴(𝑥) ≤   1 − 𝑓𝐵(𝑥)  

(ii) A ∪ B = { max( 𝑡𝐴(𝑥), 𝑡𝐵(𝑥)) ,
max( 1−𝑓𝐴(𝑥),1−𝑓𝐵(𝑥))

x
∈ X }. 

(iii) A ∩ B = { min( 𝑡𝐴(𝑥), 𝑡𝐵(𝑥)) ,
min( 1−𝑓𝐴(𝑥),1−𝑓𝐵(𝑥))

x
∈ X }. 

(iv) Ac = {𝑥, 𝑓𝐴(𝑥), 1 − 𝑡𝐴(𝑥): x ∈ X }. 

Definition 2.5: [1] Let X be the universe of discourse. A bipolar-valued vague set A in X is an object 

having the form A = {x, [𝑡𝐴
+(𝑥), 1 − 𝑓𝐴

+(𝑥)], [−1 − 𝑓𝐴
−(𝑥), 𝑡𝐴

−(𝑥)] ∶ x ∈ X } where         [𝑡𝐴
+, 1 − 𝑓𝐴

+] 

: X→[0,1] and [−1 − 𝑓𝐴
−, 𝑡𝐴

−] : X→[-1,0] are the mapping such that                            𝑡𝐴
+(𝑥) + 𝑓𝐴

+(𝑥) ≤ 

1 and -1≤ 𝑡𝐴
−+ 𝑓𝐴

−. The positive membership degree [𝑡𝐴
+(𝑥), 1 − 𝑓𝐴

+(𝑥)] denotes the satisfaction region 

of an element x to the property corresponding to a bipolar-valued set A and the negative membership 

degree [−1 − 𝑓𝐴
−(𝑥), 𝑡𝐴

−(𝑥)] denotes the satisfaction region of x to some implicit counter property of 

A. For a sake of simplicity, we shall use the notion of bipolar vague set 𝜈𝐴
+ = [𝑡𝐴

+, 1 − 𝑓𝐴
+] and 𝜈𝐴

− = 

[−1 − 𝑓𝐴
−, 𝑡𝐴

−]. 

Definition 2.6: [5] A bipolar vague set A = [𝜈𝐴
+, 𝜈𝐴

−] of a set U with 𝜈𝐴
+= 0 implies that 𝑡𝐴

+ = 0,              1 −

𝑓𝐴
+= 0 and 𝜈𝐴

−= 0 implies that 𝑡𝐴
− = 0, −1 − 𝑓𝐴

− = 0 for all x ∈ U is called zero bipolar vague set and it 

is denoted by 0. 

Definition 2.7: [5] A bipolar vague set A = [𝜈𝐴
+, 𝜈𝐴

−] of a set U with 𝜈𝐴
+= 1 implies that 𝑡𝐴

+ = 1,                        

1 − 𝑓𝐴
+= 1 and 𝜈𝐴

−= -1 implies that 𝑡𝐴
− = -1, −1 − 𝑓𝐴

− = -1 for all x ∈ U is called unit bipolar vague set 

and it is denoted by 1. 

Definition 2.8: [4]  Let A = x, [𝑡𝐴
+, 1 − 𝑓𝐴

+], [−1 − 𝑓𝐴
−, 𝑡𝐴

−] and x, [𝑡𝐵
+, 1 − 𝑓𝐵

+], [−1 − 𝑓𝐵
−, 𝑡𝐵

−] be 

two bipolar vague sets then their union, intersection and complement are defined as follows: 



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(i) A ∪ B = {x, [𝑡𝐴∪𝐵
+ (𝑥), 1 − 𝑓𝐴∪𝐵

+ (𝑥)],
[−1−𝑓𝐴∪𝐵

− (𝑥),𝑡𝐴∪𝐵
− (𝑥)]

x
∈ X } where 

          𝑡𝐴∪𝐵
+ (𝑥) = max {𝑡𝐴

+(𝑥), 𝑡𝐵
+(𝑥)}, 𝑡𝐴∪𝐵

− (𝑥) = min {𝑡𝐴
−(𝑥), 𝑡𝐵

−(𝑥)} and 

          1 − 𝑓𝐴∪𝐵
+ (𝑥) = max {1 − 𝑓𝐴

+(𝑥), 1 − 𝑓𝐵
+(𝑥)}, 

        −1 − 𝑓𝐴∪𝐵
− (𝑥) = min {−1 − 𝑓𝐴

−(𝑥), −1 − 𝑓𝐵
−(𝑥)}. 

(ii) A ∩ B = {x, [𝑡𝐴∩𝐵
+ (𝑥), 1 − 𝑓𝐴∩𝐵

+ (𝑥)],
[−1−𝑓𝐴∩𝐵

− (𝑥),𝑡𝐴∩𝐵
− (𝑥)]

x
∈ X } where 

          𝑡𝐴∩𝐵
+ (𝑥) = min {𝑡𝐴

+(𝑥), 𝑡𝐵
+(𝑥)}, 𝑡𝐴∩𝐵

− (𝑥) = max {𝑡𝐴
−(𝑥), 𝑡𝐵

−(𝑥)} and 

          1 − 𝑓𝐴∩𝐵
+ (𝑥) = min {1 − 𝑓𝐴

+(𝑥), 1 − 𝑓𝐵
+(𝑥)}, 

        −1 − 𝑓𝐴∪𝐵
− (𝑥) = max {−1 − 𝑓𝐴

−(𝑥), −1 − 𝑓𝐵
−(𝑥)}. 

(iii) Ac = {x, [𝑓𝐴
+(𝑥), 1 − 𝑡𝐴

+(𝑥)], [−1 − 𝑡𝐴
−(x), 𝑓𝐴

−(x)]/ x ∈ X}. 

Definition 2.9: [4] Let A and B be two bipolar vague sets defined over a universe of discourse X. We 

say that A ⊆ B if and only if 𝑡𝐴
+(𝑥) ≤ 𝑡𝐵

+(𝑥), 1 − 𝑓𝐴
+(𝑥) ≤ 1 − 𝑓𝐵

+(𝑥) and 𝑡𝐴
−(𝑥) ≥ 𝑡𝐵

−(𝑥),             −1 −

𝑓𝐴
−(𝑥) ≥ 1 − 𝑓𝐵

−(𝑥) for all x ∈ X. 

Definition 2.10: [4] A bipolar vague topology (BVT) on a non-empty set X is a family B𝑉𝜏 of bipolar 

vague set in X satisfying the following axioms: 

(i) 0~,1~ ∈ B𝑉𝜏 

(ii) 𝐺1 ∩ 𝐺2 ∈ B𝑉𝜏, for any 𝐺1, 𝐺2 ∈ B𝑉𝜏 

(iii) ∪ 𝐺𝑖 ∈ B𝑉𝜏, for any arbitrary family {𝐺𝑖: 𝐺𝑖 ∈ B𝑉𝜏, I ∈ I}. 

In this case the pair (X, B𝑉𝜏) is called a bipolar vague topological space and any bipolar vague set 

(BVS) in B𝑉𝜏 is known as bipolar vague open set in X. The complement Ac of a bipolar vague open set 

(BVOS) A in a bipolar vague topological space (X, B𝑉𝜏) is called a bipolar vague closed set (BVCS) 

in X. 

Definition 2.11: [4] Let (X, B𝑉𝜏) be a bipolar vague topological space                                                                              

A = x, [𝑡𝐴
+, 1 − 𝑓𝐴

+], [−1 − 𝑓𝐴
−, 𝑡𝐴

−] be a bipolar vague set in X. Then the bipolar vague interior and 

bipolar vague closure of A are defined by, 

 BVInt(A) = ∪ {G: G is a bipolar vague open set in X and G ⊆ A}, 

BVCl(A) = ∩ {K: K is a bipolar vague closed set in X and A⊆ K}. 

Note that BVCl(A) is a bipolar vague closed set and BVInt(A) is a bipolar vague open set in X. Further, 

(i) A is a bipolar vague closed set in X if and only if BVCl(A) = A, 

(ii) A is a bipolar vague open set in X if and only if BVInt(A) = A. 

Definition 2.12: [4] Let (X, B𝑉𝜏) be a bipolar vague topological space. A bipolar vague set A in          (X, 

B𝑉𝜏) is said to be a generalized bipolar vague closed set if BVCl(A) ⊆ G whenever A⊆ G and G is 

bipolar vague open. The complement of a generalized bipolar vague closed set is generalized bipolar 

vague open set. 

Definition 2.13: [4] Let (X, B𝑉𝜏) be a bipolar vague topological space and A be a bipolar vague set in 

X. Then the generalized bipolar vague closure and generalized bipolar vague interior of A are defined 

by, 



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GBVCl(A) = ∩ {G: G is a generalized bipolar vague closed set in X and A⊆ G}, 

GBInt(A) = ∪ {G: G is a generalized bipolar vague open set in X and A ⊇ G}. 

Definition 2.14: [10] A bipolar vague set A of a bipolar vague topological space X, is said to be  

(i) a bipolar vague 𝛼-open set if A ⊆ BVInt(BVCl(BVInt(A))) 

(ii) a bipolar vague pre-open set if A ⊆ BVInt(BVCl(A)) 

(iii) a bipolar vague semi-open set if A ⊆ BVCl(BVInt(A)) 

(iv) a bipolar vague semi-𝛼-open set if A ⊆ BVCl(𝛼BVInt(A)) 

(v) a bipolar vague regular-open set BVInt(BVCl(A)) = A 

(vi) a bipolar vague 𝛽-open set A ⊆ BVCl(BVInt(BVCl(A))). 

Definition 2.15: [10] A bipolar vague set A of a bipolar vague topological space X, is said to be 

(i) a bipolar vague 𝛼-closed set if BVCl(BVInt(BVCl(A))) ⊆ A 

(ii) a bipolar vague pre-closed set if BVCl(BVInt(A)) ⊆ A 

(iii) a bipolar vague semi-closed set if BVInt(BVCl(A)) ⊆ A 

(iv) a bipolar vague semi-𝛼-closed set if BVInt(𝛼BVCl(A)) ⊆ A 

(v) a bipolar vague regular-closed set if BVCl(BVInt(A)) = A 

(vi) a bipolar vague 𝛽-closed set if BVInt(BVCl(BVInt(A))) ⊆ A. 

Definition 2.16: [10] Let A be a bipolar vague set of a bipolar vague topological space (X, B𝑉𝜏). Then 

the bipolar vague 𝛼 interior and bipolar vague 𝛼 closure are defined as 

 B𝑉𝛼Int(A) = ∪ {G: G is a bipolar vague 𝛼-open set in X and G ⊆ A}, 

 B𝑉𝛼Cl(A) = ∩ {K: K is a bipolar vague 𝛼-closed set in X and A⊆ K}. 

Definition 2.17: [10] A bipolar vague set A in a bipolar vague topological space X, is said to be a 

bipolar vague 𝛼 generalized closed set if B𝑉𝛼Cl(A) ⊆ U whenever A⊆ U and U is a bipolar vague open 

set in X. The complement Ac of a bipolar vague 𝛼 generalized closed set A is a bipolar vague 𝛼 

generalized open set in X. 

Definition 2.18: [4] Let (X, B𝑉𝜏) and (Y, B𝑉𝜎) be two bipolar vague topological spaces and                        

𝑓 : X→ Y be a function. Then 𝜑 is said to be bipolar vague continuous if and only if the preimage of 

each bipolar vague open set in Y is a bipolar vague open set in X. 

Definition 2.19: [4] A map 𝑓 : (X, B𝑉𝜏) → (Y, B𝑉𝜎) is said to be generalized bipolar vague continuous 

if the inverse image of every bipolar vague open set in (Y, B𝑉𝜎) is a generalized vague open set in (X, 

B𝑉𝜏). 

Definition 2.20: [4] Let 𝑓 be a mapping from a bipolar vague topological space (X, B𝑉𝜏) into a bipolar 

vague topological space (Y, B𝑉𝜎). Then 𝑓 is said to be a bipolar vague generalized irresolute mapping 

if the inverse image of every bipolar vague generalized closed set in (Y, B𝑉𝜎) is a bipolar vague 

generalized closed set in (X, B𝑉𝜏). 

3. Bipolar Vague 𝜶 Generalized Continuous Mappings in Topological Spaces 



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In this section we have introduced bipolar vague 𝛼 generalized continuous mappings and 

investigated some of their properties. Also, we have established the relation between the newly 

introduced mappings and already existing mappings. 

Definition 3.1: Let (X, B𝑉𝜏) and (Y, B𝑉𝜎) be two bipolar vague topological spaces. Then the 

mapping 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) is called 

(i) a bipolar vague 𝛼 continuous if the inverse image of every bipolar vague closed set in (Y, B𝑉𝜎) 

is a bipolar vague 𝛼-closed set in (X, B𝑉𝜏). 

(ii)  a bipolar vague pre continuous if the inverse image of every bipolar vague closed set in (Y, 

B𝑉𝜎) is a bipolar vague pre-closed set in (X, B𝑉𝜏).  

(iii) a bipolar vague semi continuous if the inverse image of every bipolar vague closed set in (Y, 

B𝑉𝜎) is a bipolar vague semi-closed set in (X, B𝑉𝜏). 

Definition 3.2: Let (X, B𝑉𝜏) and (Y, B𝑉𝜎) be two bipolar vague topological spaces. A mapping       𝑓 : 

(X, B𝑉𝜏)  → (Y, B𝑉𝜎) is called a bipolar vague 𝛼 generalized continuous mapping if 𝑓−1(B) is a bipolar 

vague 𝛼 generalized closed set in (X, B𝑉𝜏) for every bipolar vague closed set B of             (Y, B𝑉𝜎). 

Example 3.3: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.5, 0.5] [-0.5, -0.5], [0.5, 0.5]                      [-

0.5, -0.5] and B = y, [0.7, 0.6] [-0.9, -0.9], [0.6, 0.6] [-0.5, -0.5]. Define a mapping                          𝑓 

: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Here the bipolar vague set Bc = y, [0.4, 0.3]       [-0.1, 

-0.1], [0.4, 0.4] [-0.5, -0.5] is a bipolar vague closed set in Y. Then 𝑓−1 (Bc) = x, [0.4, 0.3]       [-0.1, 

-0.1], [0.4, 0.4] [-0.5, -0.5] is a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) as            𝑓−1 (Bc) 

⊆ A and B𝑉𝛼Cl(𝑓−1 (Bc)) = 𝑓−1 (Bc) ∪ BVCl(BVInt(BVCl(𝑓−1 (Bc)))) = Ac ⊆ A, where A is a bipolar 

vague open set in X. Therefore, 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

Proposition 3.4: Every bipolar vague continuous mapping is a bipolar vague 𝛼 generalized continuous 

mapping but not conversely in general. 

Proof: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague continuous mapping. Let A be a bipolar vague 

closed set in Y. Then 𝑓−1(A) is a bipolar vague closed set in X. Since every bipolar vague closed set 

is a bipolar vague 𝛼 generalized closed set in X [10], 𝑓−1(A) is a bipolar vague 𝛼 generalized closed 

set in X. Hence 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

Example 3.5: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.2, 0.3] [-0.4, -0.4], [0.5, 0.5]                      [-

0.4, -0.4] and B = y, [0.4, 0.4] [-0.4, -0.4], [0.6, 0.6] [-0.4, -0.4]. Define a mapping                          𝑓 

: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Here the bipolar vague set Bc = y, [0.6, 0.6]       [-0.6, 

-0.6], [0.4, 0.4] [-0.6, -0.6] is a bipolar vague closed set in Y. Then 𝑓−1 (Bc) = x, [0.6, 0.6]       [-0.6, 

-0.6], [0.4, 0.4] [-0.6, -0.6] is a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) as            𝑓−1 (Bc) 

⊆ 1~  and B𝑉𝛼Cl(𝑓−1 (Bc)) = 𝑓−1 (Bc) ∪ BVCl(BVInt(BVCl(𝑓−1 (Bc)))) = Ac ⊆ 1~, where Ac is a 

bipolar vague closed set in X. Therefore, 𝑓 is a bipolar vague 𝛼 generalized continuous mapping but 

since 𝑓−1 (Bc) is not a bipolar vague closed set in X as BVCl(𝑓−1 (Bc)) = Ac ≠ 𝑓−1 (Bc), 𝑓 is not a 

bipolar vague continuous mapping. 



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Proposition 3.6: Every bipolar vague 𝛼 continuous mapping is a bipolar vague 𝛼 generalized 

continuous mapping but not conversely in general. 

Proof: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 continuous mapping. Let A be a bipolar vague 

closed set in Y. Then 𝑓−1(A) is a bipolar vague 𝛼-closed set in X. Since every bipolar vague 𝛼-closed 

set is a bipolar vague 𝛼 generalized closed set in X [10], 𝑓−1(A) is a bipolar vague 𝛼 generalized 

closed set in X. Hence 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

Example 3.7: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.2, 0.3] [-0.3, -0.3], [0.5, 0.5]                      [-

0.4, -0.4] and B = y, [0.3, 0.3] [-0.3, -0.3], [0.7, 0.7] [-0.5, -0.5]. Define a mapping                          𝑓 

: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Here the bipolar vague set Bc = y, [0.7, 0.7]       [-0.7, 

-0.7], [0.3, 0.3] [-0.5, -0.5] is a bipolar vague closed set in Y. Then 𝑓−1 (Bc) = x, [0.7, 0.7]       [-0.7, 

-0.7], [0.3, 0.3] [-0.5, -0.5] is a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) as            𝑓−1 (Bc) 

⊆ 1~  and B𝑉𝛼Cl(𝑓−1 (Bc)) = 𝑓−1 (Bc) ∪ BVCl(BVInt(BVCl(𝑓−1 (Bc)))) = Ac ⊆ 1~, where Ac is a 

bipolar vague closed set in X. Therefore, 𝑓 is a bipolar vague 𝛼 generalized continuous mapping but 

since BVCl (BVInt (BVCl(𝑓−1 (𝐵𝑐)))) = Ac ⊄ 𝑓−1 (Bc). Hence 𝑓 is not a bipolar vague 𝛼 continuous 

mapping. 

Remark 3.8: Every bipolar vague semi continuous mapping and bipolar vague 𝛼 generalized 

continuous mapping are independent to each other in general. 

Example 3.9: In Example 3.4, 𝑓 is a bipolar vague 𝛼 generalized continuous mapping but since 

BVInt (BVCl(𝑓−1 (𝐵𝑐))) = BVInt(Ac) = A ⊄ 𝑓−1 (Bc) =  x, [0.4, 0.3] [-0.1, -0.1], [0.4, 0.4]                [-

0.5, -0.5], 𝑓−1 (Bc) is not a bipolar vague semi-closed set in X. Hence 𝑓 is not a bipolar vague semi 

continuous mapping.  

Example 3.10: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.4, 0.3] [-0.2, -0.2], [0.5, 0.5]                      [-

0.5, -0.5] and B = y, [0.7, 0.6] [-0.8, -0.8], [0.5, 0.5] [-0.5, -0.5]. Define a mapping                          𝑓 

: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Here the bipolar vague set Bc = y, [0.4, 0.3]       [-0.2, 

-0.2], [0.5, 0.5] [-0.5, -0.5] is a bipolar vague closed set in Y. But 𝑓−1 (Bc) = x, [0.4, 0.3]       [-0.2, -

0.2], [0.5, 0.5] [-0.5, -0.5] is not a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) as            𝑓−1 

(Bc) ⊆ A  and B𝑉𝛼Cl(𝑓−1 (Bc)) = 𝑓−1 (Bc) ∪ BVCl(BVInt(BVCl(𝑓−1 (Bc)))) = Ac ⊄ A, where Ac is a 

bipolar vague closed set in X. Therefore, 𝑓 is not a bipolar vague 𝛼 generalized continuous mapping 

but since BVInt (BVCl(𝑓−1 (𝐵𝑐))) = A  ⊆ 𝑓−1 (Bc) is a bipolar vague semi-closed set in X. Hence 𝑓 

is a bipolar vague semi continuous mapping. 

Remark 3.11: Every bipolar vague pre continuous mapping and bipolar vague 𝛼 generalized 

continuous mapping are independent to each other in general.  

Example 3.12: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.1, 0.1] [-0.4, -0.4], [0.6, 0.3]                      [-

0.5, -0.5] and B = y, [0.2, 0.2] [-0.5, -0.5], [0.7, 0.3] [-0.5, -0.5]. Define a mapping                          𝑓 



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: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Here the bipolar vague set Bc = y, [0.8, 0.8]       [-0.5, 

-0.5], [0.7, 0.3] [-0.5, -0.5] is a bipolar vague closed set in Y. Then 𝑓−1 (Bc) = x, [0.8, 0.8]       [-0.5, 

-0.5], [0.7, 0.3] [-0.5, -0.5] is a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) as            𝑓−1 (Bc) 

⊆ 1~  and B𝑉𝛼Cl(𝑓−1 (Bc)) = 𝑓−1 (Bc) ∪ BVCl(BVInt(BVCl(𝑓−1 (Bc)))) = Ac ⊆ 1~, where Ac is a 

bipolar vague closed set in X. Therefore, 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. Since 

BVCl (BVInt(𝑓−1 (𝐵𝑐))) = Ac  ⊄ 𝑓−1 (Bc), 𝑓−1 (Bc) is not a bipolar vague pre-closed set in X. Hence 

𝑓 is not a bipolar vague pre continuous mapping. 

Example 3.13: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.5, 0.4] [-0.3, -0.2], [0.5, 0.5]                      [-

0.3, -0.2] and B = y, [0.8, 0.7] [-0.8, -0.8], [0.5, 0.5] [-0.8, -0.8]. Define a mapping                          𝑓 

: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Here the bipolar vague set Bc = y, [0.3, 0.2]       [-0.2, 

-0.2], [0.5, 0.5] [-0.2, -0.2] is a bipolar vague closed set in Y. Then 𝑓−1 (Bc) = x, [0.3, 0.2]       [-0.2, 

-0.2], [0.5, 0.5] [-0.2, -0.2] is not a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) as            𝑓−1 

(Bc) ⊆ A  and B𝑉𝛼Cl(𝑓−1 (Bc)) = 𝑓−1 (Bc) ∪ BVCl(BVInt(BVCl(𝑓−1 (Bc)))) = Ac ⊄ A, where Ac is a 

bipolar vague closed set in X. Therefore, 𝑓 is not a bipolar vague 𝛼 generalized continuous mapping. 

Since BVCl (BVInt(𝑓−1 (𝐵𝑐))) = 0~ ⊆ 𝑓−1 (Bc), 𝑓−1 (Bc) is a bipolar vague pre-closed set in X. 

Hence 𝑓 is a bipolar vague pre continuous mapping. 

The relation between various types of bipolar vague continuity is given in the following diagram: 

 

Proposition 3.14: A mapping 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) is a bipolar vague 𝛼 generalized continuous if 

and only if the inverse image of each bipolar vague open set in Y is a bipolar vague 𝛼 generalized open 

set in X. 

Proof: Necessity: Let A be a bipolar vague open set in Y. This implies Ac is a bipolar vague closed set 

in Y. Since 𝑓 is a bipolar vague 𝛼 generalized continuous, 𝑓−1(Ac) is a bipolar vague 𝛼 generalized 

closed set in X. Since 𝑓−1(Ac) =  (𝑓−1(A))c , 𝑓−1(A) is a bipolar vague 𝛼 generalized open set in X. 

 Sufficiency: Let A be a bipolar vague closed set in Y. This implies Ac is a bipolar vague open 

set in Y. By hypothesis, 𝑓−1(Ac) is a bipolar vague 𝛼 generalized open set in X. Since 𝑓−1(Ac) =  

(𝑓−1(A))c , 𝑓−1(A) is a bipolar vague 𝛼 generalized closed set in X. Hence 𝑓 is a bipolar vague 𝛼 

generalized continuous mapping. 

Bipolar Vague Continuous 

Bipolar Vague 𝜶 Generalized Continuous  

Bipolar Vague 𝛼 Continuous 

Bipolar Vague Pre Continuous Bipolar Vague Semi Continuous 



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Proposition 3.15: If 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) is a bipolar vague 𝛼 generalized continuous mapping 

and 𝑔 : (Y, B𝑉𝜎)  → (Z, B𝑉𝛿) is a bipolar vague continuous mapping, then                                                𝑔 ∘

𝑓: (X, B𝑉𝜏)  → (Z, B𝑉𝛿) is a bipolar vague 𝛼 generalized continuous mapping. 

Proof: Let A be a bipolar vague closed set in Z. Then 𝑔−1(A) be a bipolar vague closed set in Y, by 

hypothesis. Since 𝑓 is a bipolar vague 𝛼 generalized continuous mapping, 𝑓−1( 𝑔−1(A)) is a bipolar 

vague 𝛼 generalized closed set in X. Hence 𝑔 ∘ 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

Definition 3.16: Let (X, B𝑉𝜏) be a bipolar vague topological space. The bipolar vague alpha 

generalized closure (B𝑉𝛼𝑔Cl(A)) for any bipolar vague set A is defined as follows: 

B𝑉𝛼Cl(A) = ∩ {K: K is a bipolar vague 𝛼 generalized closed set in X and A⊆ K}. If A is a bipolar 

vague 𝛼 generalized closed set, then B𝑉𝛼𝑔Cl(A) = A. 

Proposition 3.17: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 generalized continuous mapping. 

Then the following conditions are hold: 

(i) 𝑓(B𝑉𝛼𝑔Cl(A)) ⊆ BVCl(𝑓(A)), for every bipolar vague set A in X. 

(ii) B𝑉𝛼𝑔Cl(𝑓−1(B)) ⊆ 𝑓−1(BVCl(B)), for every bipolar vague set B in Y. 

Proof: (i) Since BVCl(𝑓(A)) is a bipolar vague closed set in Y and 𝑓 is a bipolar vague 𝛼 generalized 

continuous mapping, then 𝑓−1(BVCl(f(A))) is a bipolar vague 𝛼 generalized closed set in X. That is 

B𝑉𝛼𝑔Cl(𝑓−1(BVCl(f(A)))) = 𝑓−1(BVCl(f(A))). Now, 𝑓(B𝑉𝛼𝑔Cl(𝑓−1(BVCl(f(A)))) = 

𝑓𝑓−1(BVCl(f(A))) ⊆ BVCl(𝑓(A)). Then 𝑓(B𝑉𝛼𝑔Cl(A)) ⊆ 𝑓(B𝑉𝛼𝑔Cl( 𝑓−1𝑓(A))) ⊆ 

𝑓(B𝑉𝛼𝑔Cl(𝑓−1(BVCl(f(A)))) ⊆ BVCl(𝑓(A)). Therefore 𝑓(B𝑉𝛼𝑔Cl(A)) ⊆ BVCl(𝑓(A)), for every 

bipolar vague set A in X. 

(ii) Replacing A by 𝑓−1(B) in (i), we get 𝑓(B𝑉𝛼𝑔Cl( 𝑓−1(B))) ⊆ BVCl(𝑓(𝑓−1(B))) ⊆ BVCl(B). Hence 

B𝑉𝛼𝑔Cl(𝑓−1(B)) ⊆ 𝑓−1(𝑓( B𝑉𝛼𝑔Cl(𝑓−1(B)))) ⊆ 𝑓−1(BVCl(B)), for every bipolar vague set B in Y. 

Definition 3.18: A bipolar vague topological space (X, B𝑉𝜏) is said to be bipolar vague                      𝛼𝑎 

𝑇1/2(B𝑉𝛼𝑎𝑇1/2) space if every bipolar vague 𝛼 generalized closed set in X is a bipolar vague closed set 

in X. 

Definition 3.19: A bipolar vague topological space (X, B𝑉𝜏) is said to be bipolar vague                      𝛼𝑏 

𝑇1/2(B𝑉𝛼𝑏𝑇1/2) space if every bipolar vague 𝛼 generalized closed set in X is a bipolar vague 

generalized closed set in X. 

Proposition 3.20: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 generalized continuous mapping, 

then 𝑓 is a bipolar vague continuous mapping, if X is a B𝑉𝛼𝑎𝑇1/2 space. 

Proof: Let A be a bipolar vague closed set in Y. Then 𝑓−1(A) is a bipolar vague 𝛼 generalized closed 

set in X, by hypothesis. Since X is a B𝑉𝛼𝑎𝑇1/2 space, 𝑓−1(A) is a bipolar vague closed set in X. Hence 

𝑓 is a bipolar vague continuous mapping.   

Proposition 3.21: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 generalized continuous mapping, 

then 𝑓 is a bipolar vague generalized continuous mapping, if X is a B𝑉𝛼𝑏𝑇1/2 space. 



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Proof: Let A be a bipolar vague closed set in Y. Then 𝑓−1(A) is a bipolar vague 𝛼 generalized closed 

set in X, by hypothesis. Since X is a B𝑉𝛼𝑏𝑇1/2 space, 𝑓−1(A) is a bipolar vague generalized closed set 

in X. Hence 𝑓 is a bipolar vague generalized continuous mapping.   

Proposition 3.22: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a mapping from a bipolar vague topological space 

X into a bipolar vague topological space Y. Then the following conditions are equivalent if X is a 

B𝑉𝛼𝑎𝑇1/2 space: 

(i) 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

(ii) If B is a bipolar vague open set in Y, 𝑓−1(B) is a bipolar vague 𝛼 generalized closed set in X. 

(iii) 𝑓−1(BVInt(B)) ⊆ BVInt(BVCl(BVInt(𝑓−1(B)))) for every bipolar vague set B       in Y. 

Proof: (i) ⟹ (ii) is obviously true. 

(ii) ⟹ (iii). Let B be any bipolar vague open set in Y. The BVInt(B) is a bipolar vague open set in Y. 

Then 𝑓−1(BVInt(B)) is a bipolar vague 𝛼 generalized open set in X. Since X is a B𝑉𝛼𝑎𝑇1/2 space, 

𝑓−1(BVInt(B)) is a bipolar vague open set in X. Therefore, 𝑓−1(BVInt(B)) = BVInt(𝑓−1(BVInt(B))) 

⊆ BVInt(BVCl(BVInt(𝑓−1(B)))). 

 (iii) ⟹ (i). Let B be a bipolar vague closed set in Y. Then its complement Bc is a bipolar vague open 

set in Y. By hypothesis, 𝑓−1(BVInt(Bc)) ⊆ BVInt(BVCl(BVInt(𝑓−1(Bc)))). This implies  𝑓−1( Bc) ⊆ 

BVInt(BVCl(BVInt(𝑓−1(Bc)))). Hence 𝑓−1( Bc) is a bipolar vague 𝛼-open set in X. Since every bipolar 

vague 𝛼-open set is a bipolar vague 𝛼 generalized open set, 𝑓−1( Bc) is a bipolar vague 𝛼 generalized 

open set in X. Therefore, 𝑓−1(B) is a bipolar vague 𝛼 generalized closed set in X. Hence 𝑓 is a bipolar 

vague 𝛼 generalized continuous mapping. 

Proposition 3.23: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a mapping from a bipolar vague topological space 

X into a bipolar vague topological space Y. Then the following conditions are equivalent if X is a 

B𝑉𝛼𝑎𝑇1/2 space: 

(i) 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

(ii) If 𝑓−1(B) is a bipolar vague 𝛼 generalized closed set in X, for every bipolar vague closed set 

B in Y. 

(iii) BVCl(BVInt(BVCl(𝑓−1(A)))) ⊆  𝑓−1(BVCl(A)) for every bipolar vague set B       in Y. 

Proof: (i) ⟹ (ii) is obviously true. 

(ii) ⟹ (iii). Let A be any bipolar vague set in Y. Then BVCl(A) is a bipolar vague closed set in Y. By 

hypothesis, 𝑓−1(BVCl(A)) is a bipolar vague 𝛼 generalized closed set in X. Since X is a B𝑉𝛼𝑎𝑇1/2 

space, 𝑓−1(BVCl(A)) is a bipolar vague closed set in X. Therefore,                                   

BVCl(𝑓−1(BVCl(A))) = 𝑓−1(BVCl(A)). Now BVCl(BVInt(BVCl((𝑓−1(A)))) ⊆ 

BVCl(BVInt(BVCl((𝑓−1(BVCl(A)))) ⊆  𝑓−1(BVCl(A)). 

(iii) ⟹ (i). Let A be a bipolar vague closed set in Y. Then by hypothesis, 

BVCl(BVInt(BVCl(𝑓−1(BVCl(A))))) ⊆  𝑓−1(BVCl(A)) = 𝑓−1(A). This implies 𝑓−1(A) is a bipolar 

vague 𝛼-closed set in X and hence it is a bipolar vague 𝛼 generalized closed set in X. Therefore, 𝑓 is 

a bipolar vague 𝛼 generalized continuous mapping. 



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4. Bipolar Vague 𝜶 Generalized Irresolute Mappings in Topological Spaces  

 In this section we have introduced bipolar vague 𝛼 generalized irresolute mappings and studied 

some of their properties. 

Definition 4.1:  A mapping 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) is called a bipolar vague 𝛼 generalized irresolute 

mapping if 𝑓−1(A) is a bipolar vague 𝛼 generalized closed set in (X, B𝑉𝜏) for every bipolar vague 𝛼 

generalized closed set A of (Y, B𝑉𝜎). 

Proposition 4.2: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 generalized irresolute mapping, 

then 𝑓 is a bipolar vague 𝛼 generalized continuous mapping but not conversely. 

Proof: Let 𝑓 be a bipolar vague 𝛼 generalized irresolute mapping. Let A be any bipolar vague closed 

set in Y. Since every bipolar vague closed set is a bipolar vague 𝛼 generalized closed             set [10], 

A is a bipolar vague 𝛼 generalized closed set in Y. By hypothesis, 𝑓−1(A) is a bipolar vague 𝛼 

generalized closed set in X. Hence 𝑓 is a bipolar vague 𝛼 generalized continuous mapping. 

Example 4.3: Let X = {a,b} and Y = {u,v}. Then 𝜏 = {0~, A, 1~}  and 𝜎 = {0~, B, 1~} are bipolar 

vague topologies on X and Y respectively, where A = x, [0.1, 0.3] [-0.3, -0.3], [0.6, 0.3]                      [-

0.3, -0.3] and B = y, [0.3, 0.1] [-0.1, -0.1], [0.5, 0.6] [-0.1, -0.1]. Define a mapping                          𝑓 

: (X, B𝑉𝜏)  → (Y, B𝑉𝜎) by f(a) = u and f(b) = v. Then 𝑓 is a bipolar vague 𝛼 generalized continuous 

mapping but not a bipolar vague 𝛼 generalized irresolute mapping. Since the bipolar vague set     M = 

y, [0.1, 0.3] [-0.2, -0.2], [0.6, 0.2] [-0.2, -0.2] is a bipolar vague 𝛼 generalized closed set in Y but  

𝑓−1 (M) is not a bipolar vague 𝛼 generalized closed set in X as 𝑓−1 (M) =  x, [0.1, 0.3]      [-0.2, -

0.2], [0.6, 0.2] [-0.2, -0.2] ⊆ A but B𝑉𝛼Cl(𝑓−1 (M)) = 𝑓−1 (M) ∪ BVCl(BVInt(BVCl(     𝑓−1 (M)))) 

= Ac ⊄ A. Hence 𝑓 is not a bipolar vague 𝛼 generalized irresolute mapping. 

Proposition 4.4: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) and 𝑔 : (Y, B𝑉𝜎) → (Z, B𝑉𝛿) be any two bipolar vague 

𝛼 generalized irresolute mappings, then 𝑔 ∘ 𝑓 : (X, B𝑉𝜏)  → (Z, B𝑉𝛿) is a bipolar vague 𝛼 generalized 

irresolute mapping. 

Proof: Let A be a bipolar vague 𝛼 generalized closed set in Z. Then 𝑔−1 (A) is a bipolar vague 𝛼 

generalized closed set in Y. Since 𝑓 is a bipolar vague 𝛼 generalized irresolute mapping,      𝑓−1(𝑔−1 

(A)) is a bipolar vague 𝛼 generalized closed set in X. Hence 𝑔 ∘ 𝑓 is a bipolar vague 𝛼 generalized 

irresolute mapping. 

Proposition 4.5: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 generalized irresolute mapping and 

𝑔 : (Y, B𝑉𝜎) → (Z, B𝑉𝛿) be a bipolar vague 𝛼 generalized continuous mapping, then                   𝑔 ∘ 𝑓 

: (X, B𝑉𝜏)  → (Z, B𝑉𝛿) is a bipolar vague 𝛼 generalized continuous mapping. 

Proof: Let A be a bipolar vague closed set in Z. Then 𝑔−1 (A) is a bipolar vague 𝛼 generalized closed 

set in Y, by hypothesis. Since f is a bipolar vague 𝛼 generalized irresolute mapping,  𝑓−1(𝑔−1 (A)) is 

a bipolar vague 𝛼 generalized closed set in X. Hence 𝑔 ∘ 𝑓 is a bipolar vague 𝛼 generalized continuous 

mapping. 

Proposition 4.6: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a bipolar vague 𝛼 generalized irresolute mapping and 

𝑔 : (Y, B𝑉𝜎) → (Z, B𝑉𝛿) be a bipolar vague continuous mapping, then                                                    𝑔 ∘ 𝑓 

: (X, B𝑉𝜏)  → (Z, B𝑉𝛿) is a bipolar vague 𝛼 generalized continuous mapping.   



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Proof: Let A be a bipolar vague closed set in Z. Then 𝑔−1 (A) is a bipolar vague closed set in Y. Since 

every bipolar vague closed set is a bipolar vague 𝛼 generalized closed set [10], 𝑔−1 (A) is a bipolar 

vague 𝛼 generalized closed set in Y. Therefore  𝑓−1(𝑔−1 (A)) is a bipolar vague 𝛼 generalized closed 

set in X, by hypothesis. Hence 𝑔 ∘ 𝑓 is a bipolar vague 𝛼 generalized continuous mapping.   

 

 

Proposition 4.7: A mapping 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) is a bipolar vague 𝛼 generalized irresolute 

mapping if and only if the inverse image of each bipolar vague 𝛼 generalized open set in Y is a bipolar 

vague 𝛼 generalized open set in X. 

Proof: Necessity: Let A be a bipolar vague 𝛼 generalized open set in Y. Then Ac is a bipolar vague 𝛼 

generalized closed set in Y. Since 𝑓 is a bipolar vague 𝛼 generalized irresolute, 𝑓−1(Ac) is a bipolar 

vague 𝛼 generalized closed set in X. Since 𝑓−1(Ac) = (𝑓−1(A))c, 𝑓−1 (A) is a bipolar vague 𝛼 

generalized open set in X. 

 Sufficiency: Let A be a bipolar vague 𝛼 generalized closed set in Y. This implies Ac is a bipolar 

vague 𝛼 generalized open set in Y. By hypothesis, 𝑓−1(Ac) is a bipolar vague 𝛼 generalized open set 

in X. Since 𝑓−1(Ac) = (𝑓−1(A))c, 𝑓−1 (A) is a bipolar vague 𝛼 generalized closed set in X. Hence 𝑓 is 

a bipolar vague 𝛼 generalized irresolute mapping. 

Proposition 4.8: Let 𝑓 : (X, B𝑉𝜏)  → (Y, B𝑉𝜎) be a mapping from a bipolar vague topological space X 

into a bipolar vague topological space Y. Then the following conditions are equivalent if X and Y are 

B𝑉𝛼𝑎𝑇1/2 spaces: 

(i) 𝑓 is a bipolar vague 𝛼 generalized irresolute mapping. 

(ii) 𝑓−1 (B) is a bipolar vague 𝛼 generalized open set in X for each bipolar vague 𝛼 generalized 

open set in Y. 

(iii) BVCl(𝑓−1 (B)) ⊆  𝑓−1(BVCl(B)) for each bipolar vague set B of Y. 

Proof: (i) ⟹ (ii) is obviously true from the Proposition 4.7. 

(ii) ⟹ (iii). Let B be any bipolar vague set in Y and B ⊆ BVCl(B). Then 𝑓−1 (B) ⊆  𝑓−1(BVCl(B)). 

Since BVCl(B) is a bipolar vague closed set in Y, 𝑓−1(BVCl(B)) is a bipolar vague 𝛼 generalized 

closed set in X, by hypothesis. Since X is a B𝑉𝛼𝑎𝑇1/2 space, 𝑓−1(BVCl(B)) is a bipolar vague closed 

set in X. Hence BVCl(𝑓−1 (B)) ⊆  BVCl(𝑓−1(BVCl(B))) = 𝑓−1(BVCl(B)). 

(iii) ⟹ (i). Let B be a bipolar vague 𝛼 generalized closed set in Y. Since Y is a B𝑉𝛼𝑎𝑇1/2 space,    B is 

a bipolar vague closed set in Y and BVCl(B) = B. Hence 𝑓−1 (B) = 𝑓−1(BVCl(B)) ⊇             BVCl(𝑓−1 

(B)). But 𝑓−1 (B) ⊆ BVCl(𝑓−1 (B)). Therefore, BVCl(𝑓−1 (B)) = 𝑓−1 (B). This implies 𝑓−1 (B) is a 

bipolar vague closed set and hence it is a bipolar vague 𝛼 generalized closed set in X. Thus 𝑓 is a 

bipolar vague 𝛼 generalized irresolute mapping. 

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